Orifice plate flowmeters

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Merryn Horrocks

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Merryn Horrocks 1 year, 2 months ago

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Don Shearman 3 years, 8 months ago

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Question statement

In the design of orifice plate flowmeters, the volumetric flowrate, $Q$ (m $^3$ s $^{-1}$ ), is given by

$Q=C_dA_o\sqrt{\frac{2gh}{1-A_o^2/A_p^2}}$ where

$C_d$ is a dimensionless discharge coefficient,

$h$ (m) is the head difference across the orifice plate,

$A_o$ (m

$^2$ ) is the area of the orifice and

$A_p$ (m

$^2$ ) is the area of the pipe.

Give any introductory information the student needs.

			Name	Type	Generated Value

Q	integer	140
A_p	integer	10
C_d	number	0.9
h	integer	600
answer	number	0.7597691253
Q_eval	rational	7/50000
Apr_eval	rational	1/20
Ap_eval	number	0.0078539816
h_eval	rational	3/5
den_under_root_term1	string	5.88186e-4
den_under_root	string	5.88206e-4
numerator	string	1.09956e-6
denominator	string	2.42530e-2
test	string	k
Ao_eval	string	4.53370e-5
Ao_eval_on_pi	string	1.44312e-5
root_Ao_eval_on_pi	decimal	dec("0.0037988")

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Value

xxxxxxxxxx
 
random(60..150 #10) 

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Description

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Generated value: `integer`

140

→ Used by:

Q_eval

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Mathematical expression
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Part b) Gap-fill
1. Gap Number entry
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Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Rearrange the equation to solve for the area of the orifice, $A_o$ , in terms of the other variables. (Note: to enter the variable name $A_1$ type A_1, for $\sqrt{a/b}$ type sqrt(a/b).)

$A_o=$ Unnamed gap

Advice

Part (a)

$\begin{align}Q&=C_dA_o\sqrt{\frac{2gh}{1-A_o^2/A_p^2}}\\ \frac{Q}{C_dA_o}&=\sqrt{\frac{2gh}{1-A_o^2/A_p^2}}\\ \left(\frac{Q}{C_dA_o}\right)^2&=\frac{2gh}{1-A_o^2/A_p^2}\\ \left(\frac{C_dA_o}{Q}\right)^2&=\frac{1-A_o^2/A_p^2}{2gh}\\ \left(\frac{C_d}{Q}\right)^2A_o^2&=\frac{1}{2gh}-\frac{A_o^2}{2ghA_p^2}\\ \left(\frac{C_d}{Q}\right)^2A_o^2 + \frac{A_o^2}{2ghA_p^2} &=\frac{1}{2gh}\\ \left[\frac{C_d^2}{Q^2} + \frac{1}{2ghA_p^2}\right]A_o^2 &=\frac{1}{2gh}\\ \left[\frac{C_d^22ghA_p^2+Q^2}{Q^22ghA_p^2}\right]A_o^2 &=\frac{1}{2gh}\\ A_o^2&=\frac{1}{2gh}\times\frac{Q^22ghA_p^2}{C_d^22ghA_p^2+Q^2}\\A_o&=\frac{QA_p}{\sqrt{C_d^22ghA_p^2+Q^2}}\end{align}$

Part(b)

$\sqrt{\var{Ao_eval_on_pi}}$

To begin with, we need to work out the values of each of the variables in the correct units for the equation.

$Q = \var{Q}$ cm $^3$ s $^{-1}$ = $\var{Q}\div 1000000$ m $^3$ s $^{-1}= \var{Q_eval}$ m $^3$ s $^{-1}$

Pipe diameter = $\var{A_p}$ cm = $\var{A_p} \div 100$ m = $\var{2*Apr_eval}$ m

Pipe radius = Pipe diameter $\div 2 = \var{2*Apr_eval}\div 2 = \var{Apr_eval}$ m

A $_p = \pi\times$ pipe radius $^2 = \pi\times \var{Apr_eval}^2 = \var{Ap_eval}$ m $^2$

C_d = $0.8$

g = $9.81$ ms $^{-2}$

h = $\var{h}$ mm = $\var{h}\div 1000$ m = $\var{h_eval}$ m

So:

$\begin{align}A_o&=\frac{QA_p}{\sqrt{C_d^22ghA_p^2+Q^2}}\\ &= \frac{\var{Q_eval}\times\var{dpformat(Ap_eval,5,"scientific")}}{\sqrt{(\var{C_d})^2\times 2\times 9.81 \times \var{h_eval}\times (\var{dpformat(Ap_eval,5,"scientific")})^2+(\var{Q_eval})^2}} \\ &= \frac{\var{numerator}}{\sqrt{\var{den_under_root_term1}+\var{dpformat(Q_eval*Q_eval,5,"scientific")}}} \\ &= \frac{\var{numerator}}{\sqrt{\var{den_under_root}}}\\ &= \frac{\var{numerator}}{\var{denominator}}\\ &=\var{Ao_eval}\text{ m}\\ \text{And } A_o &= \pi \times \text{ radius}^2\\ \var{Ao_eval} & = \pi\times \text{ radius}^2\\ \text{radius}^2 & =\frac{\var{Ao_eval}}{\pi} \\ & = \var{Ao_eval_on_pi}\\ \text{radius}&=\sqrt{\var{Ao_eval_on_pi}}\\ \text{radius}&=\var{root_Ao_eval_on_pi} \text{ m}\\ & =\var{root_Ao_eval_on_pi}\times 100 \text{ cm}\\ & =\var{root_Ao_eval_on_pi*100} \text{ cm}\\ \text{Since diameter }&=\text{ radius }\times 2\\ \text{diameter } &= \var{root_Ao_eval_on_pi*100}\times 2 \text{ cm}\\ \text{diameter } &= \var{precround(root_Ao_eval_on_pi*100*2,3)} \text{ cm}\\ \end{align}$

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This question is used in the following exams:

Formulae and Transposition by Don Shearman in HELM books.
HELM Book 1.5 exercises by Merryn Horrocks in HELM books.

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Merryn Horrocks 1 year, 2 months ago

Merryn Horrocks 1 year, 2 months ago

Merryn Horrocks 1 year, 2 months ago

Don Shearman 3 years, 8 months ago

Error in variable testing condition

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Generated value: `integer`

Parts

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Orifice plate flowmeters

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Merryn Horrocks 1 year, 2 months ago

Merryn Horrocks 1 year, 2 months ago

Merryn Horrocks 1 year, 2 months ago

Don Shearman 3 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Part (a)

Part(b)

Generated value: `integer`