Let us consider that we have experimental data comprising the values of two variables x and y. We need to find a possible relationship between these two variables. The method of obtaining the specific relation in the form
y = f (x)
for the given set of experimental data to satisfy as accurately as possible is called curve fitting. This technique is generally used for the verification of experimental results.
EXAMPLE 1.24 Fit a straight line
y = ax + b
in the least square sense for the data
x 1 2 3 6 4 5
y 1 3 5 7 9 11
Solution: The normal equations for curve fitting
y = ax + b………(i) are given by
y = a x + nb………(ii)
xy = a x2 + b x………(iii) We construct the table based on the requirement. _
x y xy x2
1 1 1 1
2 3 6 4
6 7 42 36 4 9 36 16 5 11 55 25 S x = 21 S y = 36 S xy = 155 S x2 = 91 n = 6 21a + 6b = 36………(iv) 91a + 21b = 155………(v)
Multiplying Eq. (iv) by 21 and Eq. (v) by 6, we get
Substituting Eqs. (viii) and (ix) in Eq. (i), we get
y = 1.6571x + 0.2
This is the best fit of a straight line.
1.14
Regression
Let us consider that we have experimental data comprising the values of two variables x and y. We need to find the possible relationship for y in terms of x or x in terms of y. The method of obtaining the specific relation of one independent variable in terms of the other in the form
is called Regression where y = Dependent variable x = Independent variable r = Regression s = Standard deviation
n = Number of set of data.
Exercises
1.1 Discuss the fundamental and derived units with suitable examples. 1.2 What is dimensional consistency?
1.3 What is dimensional consistency equation? Explain with suitable example. 1.4 What is dimensionless group? Explain with suitable example.
1.5 One hundred pounds of water is flowing through a pipe at the rate of 10 ft/s. What is the kinetic
energy of this water in joules?
1.6 The volumetric flow rate of kerosene in 80 mm nominal diameter pipe is 75 Imperial gallons per
minute. Density of kerosene is 0.8 g/cm3. Find the mass flow rate in kg/h.
1.7 Iron metal weighing 500 lb occupies a volume of 29.25 l. Calculate the density of Fe in kg/m3. 1.8 The diameter and height of a vertical cylindrical tank are 5 ft and 6.5 ft respectively. It is full up
to 75% height with carbon tetrachloride (CCl4), the density of which is 1.6 kg/l. Find the mass in kilogram.
1.9 Show that Reynolds number defined as
is dimensionless where
D = Diameter of the pipe (m)
V = Average velocity of fluid (m/s) r = Density of the fluid (kg/m3) m = Viscosity of the fluid (kg/m s)
1.10 Find the mass flow rate of the liquid in lb/min flowing through a pipe of 5 cm diameter which
has density of 960 kg/m3 and viscosity 0.9 centipose. The Reynolds number is reported to be 3200 for the flow. [VTU Exam, Dec 2009]
1.11 Steam is flowing at the rate of 2000 kg/h in a 3 NB, 40 schedule pipe at 440 kPa absolute and
453 K. Calculate the velocity of the steam in the pipeline.
Data: Internal diameter of 3 NB, 40 schedule pipe = 3.068. Specific volume of steam at 440 kPa and
1.12 The conductance of a fluid flow system is defined as the volumetric flow rate referred to a
pressure of one torr. For an orifice, the conductance C can be computed from
where
A = Area of opening (ft2) T = Temperature (°F) M = Molecular weight.
Convert the empirical equation into SI units.
1.13 A handbook shows that microchip etching roughly follows the relation
d = 16.2 – 16.2 e –0.021t for t > 200
where
d = Depth of etch (m) t = Time of etch (second)
What are the units associated with the number 16.2 and 0.021?
Convert the relation so that d can be expressed in inches and ‘t’ can be used in minutes.
1.14 The conductance of a fluid flow system is defined as the volumetric flow rate referred to a
pressure of one torr. For an orifice, the conductance C can be computed from
where
A = Area of opening (ft2) T = Temperature (°F) M = Molecular weight
Convert the empirical equation into metric units.
1.15 What are the advantages of SI system of units? 1.16 Convert 23.16 lb ft/min2 to kg cm/s2.
1.17 Covnert 120 hp to kJ/min. 1.18 Convert
1.19 A quantity k depends on temperature T in the following manner:
where, units of quantity 20,000 is cal/mol and temperature T in K. What are the units of the value 1.2 × 105 and 1.987?
1.20 Is the following equation dimensionally homogeneous? where DP = Pressure drop (lb/ft2) L = Pipe length (ft) = Fluid velocity (ft/s) m = Fluid viscosity (lb/ft s) D = Pipe diameter (ft)
If so, are the units consistent? If not, what factor must be added to the right hand side of the equation to provide consistency.
1.21 Convert 40 l/(m2)(h) to m3/(cm2)(s). 1.22 The thermal conductivity of steel is
16 Btu/(h)(ft)(°F)
What is the value of the thermal conductivity in W/m °C?
1.23 The heat transfer coefficient for the gas flowing over a solid surface is calculated by the
empirical equation
h = 0.01G0.8
where
h = Heat transfer coefficient (Btu/(h)(ft2)(°F)) G = Mass velocity (lb/(hr)(ft2))
Convert this equation to suit SI units.
1.24 Convert 1 kWh to Btu. 1.25 Convert – 40 °C to °F.
1.26 The equation for the heat transfer from a stream of gas flowing in turbulent motion is as follows
where
Cp = Heat capacity (Btu/(lb)(°F)) G = Mass velocity (lb/(ft2) (s)) D = Internal diameter of pipe (m)
Convert this equation into SI units and find out the new constant.
1.27 In case of liquids, the local heat transfer coefficient for long tube is expressed by the empirical
where
h = Heat transfer coefficient (Btu/(h)(ft2)°F) G = Mass velocity (lb/(ft2)(s))
Cp = Heat capacity (Btu/(lb)(°F))
k = Thermal conductivity (Btu/(h)(ft)(°F)) D = Diameter of the tube (ft).
m = Viscosity of the liquid (lb/(ft) (s))
Convert the empirical equation to suit SI system of units.
1.28 Convert a volumetric flow rate of 2m3/s to l/s.
1.29 In double effect evaporator plant, the second effect is maintained under vacuum of 475 torr
(mmHg). Find the absolute pressure in kPa.
1.30 A force equal to 19.635 kgf is applied on the piston with a diameter of 5 cm. Find pressure
exerted on a piston in kPa.
1.31 Convert the pressure of 2 atm into mmHg. 1.32 Convert 2000 W in hp and (kgf m)/s. 1.33 Convert 1000 dyne into Newton. 1.34 Convert 1500 mmHg into atm. 1.35 Convert 130 lb/ft3 into g/cm3. 1.36 Make the following conversions:
(i) 350 l per minute to m3/s (ii) 475 mmHg to kN/m2.
1.37 The equation for the heat transfer to or from a stream of gas flowing in turbulent motion is as
follows:
where
Cp = Heat capacity as (Btu/lbm°F) D = Internal diameter of pipe (in) G = Mass velocity (lb/ft2 s)
h = Heat transfer co-efficient (Btu/(h)(ft2) (°F))
It is desired to transform the equation into a new form
where
Cp = Heat capacity (kcal/(kg)(°C)) D = Internal diameter of pipe (cm)
G = Mass velocity (kg/(m2)(s)) h = Heat transfer co-efficient (kcal/(h)(m2)(°C))
1.38 Prove the following:
(i) 1 g/cm3 = 62.42 lb/ft3
(ii) 1 Btu/(lb)(°F) = 1 kcal/(kg)(°C)
1.39 Prove the following:
(i) 1 Bar = 750 mmHg (ii) – 40 °C = – 40 °F
1.40 An empirical equation for calculating the inside heat transfer coefficient hi for the turbulent flow
of liquids in a pipe is given by
where
hi = Heat transfer coefficient (Btu/(h)(ft2)(°F)) G = Mass velocity of the liquid (lbm/(ft2)(h))
k = Thermal conductivity of the liquid (Btu/(h)(ft)(°F)) Cp = Heat capacity of the liquid (Btu/(lbm)(°F))
m = Viscosity of the liquid (lbm/(ft)(h)) D = Inside diameter of the pipe (ft)
(i) Verify whether the equation is dimensionally consistent or not.
(ii) What will be the value of the constant given as 0.023, if all the variables in the equation are inserted in SI units and hi is in SI units.
1.41 Iron metal weighing 500 lb occupies a volume of 29.25 ft3. Calculate the density of Fe in g/cm3. 1.42 The diameter and height of a vertical cylindrical tank are 5 ft and 6.5 ft respectively. It is full up
to 75% height with carbon tetrachloride, the density of which is 1.6 g/cm3. Find the mass in tonnes. [Ans. 4.34t]
1.43 Corrosion rate are normally reported in miles per year (mpy) in the chemical process industry.
For the measurement of the rate, a corrosion test coupon is inserted in the process stream for a definite period. The loss of weight is measured during the period of insertion.
In a particular test, a coupon of carbon steel was kept in a cooling water circuit. The dimensions of the coupon were measured to be 7.5 cm × 1.3 cm × 0.15 cm. Weight of the coupon before insertion in the circuit and after exposure for 50 days were measured to be 15 g and 14.6 g respectively. Calculate the rate of corrosion. Take the density of the carbon steel = 7.754 g/cm3.
Note: 1 mpy = 0.001 in per year. [Ans. 5.3 mpy]
1.44 Show that Prandtl number defined as
where
NPr = Prandtl number
Cp = Specific heat (J/kg °C)
m = Viscosity of the fluid (kg/m s)
k = Thermal conductivity of fluid (W/m °C)
1.45 Show that Dittus number defined as
is dimensionless where
NDi = Dittus number
h = Heat transfer coefficient d = Diameter of the pipe
k = Thermal conductivity of the fluid.
1.46 What are the different system of units in common use? With a suitable example, explain how a
quantity of one system be converted into other system.
1.47 Make the following conversions
(i) 325 l/min to m3/s (ii) 450 mmHg to kN/m2
1.48 When a gas is flowing in a tube and the tube is heated from outside. The heat transfer coefficient
is found to be related by the following equation:
h = 120 (1 + 2V1/2)
where
h = Heat transfer coefficient (Btu/h ft2 °F) V = Velocity (ft/s)
If so convert the relation for other units.
1.49 What are the limitations of FPS system? Explain with example.
1.50 Define the following terms and mention their dimensions in all the four unit system.
(i) Acceleration due to gravity (ii) Heat
(iii) Pressure (iv) Velocity
1.51 Prove the following:
(i) 1 g/cm3 = 62.42 lb/ft3 (ii) 1 Btu/lb °F = 1 kcal/kg °C
1.52 What are the fundamental quantities in FPS, CGS, MKS and SI units? 1.53 What is a derived quantity? Establish relation between Dyne and Newton.
Cp = 6.34 + 10.15 × 10–3 T – 3.41 × 10–6 T2
where, Cp is in cal/gmol °C and T is in K.
Transform the above equation in FPS unit on mass basis.
1.55 The value of heat transfer coefficient in a particular heat exchange operation is found to be
0.125 kcal/(s)(m2)(°C) Convert its value in Btu/h ft2 °C
1.56 Convert 2 g/cm3 into lb/ft3.
1.57 The value of heat transfer coefficient in a particular heat exchange operation is given by 350
W/m2 °C. Convert its value in Btu/h ft2 °C.
1.58 The superficial mass velocity of air through a dehumidifier is found to be 250 lb/(h)
(ft2).Calculate its equivalent value in kg/(h)(m2).
1.59 The superficial mass velocity of air through a dehumidifier is found to be 2000 kg/(h)
(m2).Convert its value in lb/(h)(ft2).
1.60 The variation of heat capacity data for gaseous SO2 is given by the following equation:
Cp = 43.46 + 10.64 × 10–3 T –
where Cp = cal/(gmol)(°C) and T is in K.
Transform the above equation in FPS units on mole basis.
1.61 The variation of heat capacity data for gaseous N2 is given by the following equation:
Cp = 29.49 – 5.14 × 10–3 T + 13.18 × 10–6 T2 – 4.95 × 10–9 T3
where Cp is in kJ/kmol K and T is in K. Transform the above equation in FPS units on mole basis.
1.62 At 350 °C the heat transfer by conduction and convection of a spherical mild steel ball to air is
reported to be 16 W/m2 °C. Convert this result into FPS unit.
1.63 The variation of heat capacity data for toluene is given by the equation
Cp = 1.8 + 812.21 × 10–3 T – 1512.67 × 10–6 T2
where Cp is in kJ/kmol K and T is in K. Transform the equation in FPS unit on mole basis.
1.64 Convert 390 Btu/(ft2)(h) °F to kcal/(m2)(h) °C. 1.65 Prove that
1.66 The variation of heat capacity data for O2 is given by the equation
where Cp is in kJ/kmol K and T is in K.
Transform the equation in FPS unit on mole basis.
1.67 A flow system has been found to obey
f (Q, H, g, V0, f) = 0
where
Q = Volumetric flow rate H = Liquid level
g = Acceleration due to gravity V0 = Velocity of approach f = Angle of the flow meter
Derive an equation for Q using dimensional analysis.
1.68 Explain Rayleigh’s method of dimensional analysis.
1.69 Using Buckingham’s p-theorem show that the velocity through a circular orifice is given by
where
V = Velocity of fluid flowing H = Head causing flow
D = Diameter of orifice
m = Viscosity of fluid flowing r = Density of fluid flowing g = Acceleration due to gravity
1.70 State the Buckingham’s p-theorem and explain the procedure to solve the problem for resistance
for R of a supersonic plane during flight which can be considered as dependent upon the length of the aircraft l, velocity V, air viscosity m, air density r and bulk modulus of air k. Express the functional relationship between these variables and the resistance force.
1.71 The pressure drop DP in a pipe of diameter D and length l due to turbulent flow depends on the
velocity V, viscosity m, density r and roughness k. Using Buckingham’s p-theorem, obtain an expression for DP.
1.72 Show by Rayleigh’s method of dimensional analysis that the resistance R to the motion of sphere
of diameter D moving with a uniform velocity V through a real fluid having density r and viscosity