The Prandtl number, NPr, is a dimensionless group important in heat-transfer calculations. It is defined as Cp/k, where Cp is the heat capacity of a fluid,  is the fluid viscosity, and k is the thermal conductivity. For a particular fluid, Cp = 0.583 J/(goC), k = 0.286 W/(moC), and  = 1936 lbm/(ft-h). Estimate the value of NPr without using a calculator (remember, it is dimensionless), showing your calculations; then determine it with a calculator. Express your answer in scientific notation making sure it has the correct number of significant figures.

Question

The Prandtl number, NPr, is a dimensionless group important in heat-transfer calculations. It is defined as Cp/k, where Cp is the heat capacity of a fluid,  is the fluid viscosity, and k is the thermal conductivity. For a particular fluid, Cp = 0.583 J/(goC), k = 0.286 W/(moC), and  = 1936 lbm/(ft-h). Estimate the value of NPr without using a calculator (remember, it is dimensionless), showing your calculations; then determine it with a calculator. Express your answer in scientific notation making sure it has the correct number of significant figures.

Answer 1

To estimate the value of NPr without using a calculator, we can simplify the calculation by converting the given units to suitable units for the equation.

First, convert Cp from J/(goC) to J/(kgK) since the units for mass and temperature are more commonly used. We know that 1 goC is equal to 0.001 kgK, so:

Cp = 0.583 J/(goC) * 0.001 kgK/(1 goC) = 0.000583 J/(kgK)

Next, convert k from W/(moC) to W/(kgK) since we want the units of thermal conductivity to match the units of heat capacity. We know that 1 moC is equal to 0.001 kgK, so:

k = 0.286 W/(moC) * 0.001 kgK/(1 moC) = 0.000286 W/(kgK)

Finally, convert  from lbm/(ft-h) to kg/(m s) since we want the units of viscosity to match the units of heat capacity and thermal conductivity. We know that 1 lbm is equal to 0.453592 kg, 1 ft is equal to 0.3048 m, and 1 h is equal to 3600 s, so:

 = 1936 lbm/(ft-h) * 0.453592 kg/(1 lbm) * 1/(0.3048 m/1 ft) * 1/(3600 s/1 h) = 0.468176 kg/(m s)

Now we can calculate the dimensionless Prandtl number NPr:

NPr = Cp/k = (0.000583 J/(kgK))(0.468176 kg/(m s))/(0.000286 W/(kgK)) = 1.8934014

To determine the value of NPr with a calculator, we can use the given values directly in the equation:

NPr = (0.583 J/(goC))(1936 lbm/(ft-h))/(0.286 W/(moC)) ≈ 1.8934

Therefore, the value of NPr, using a calculator, is approximately 1.8934.

Expressing the answer in scientific notation with the correct number of significant figures, we get:

NPr ≈ 1.9 x 10^0

Answer 2

To estimate the value of NPr without using a calculator, we can make approximate calculations:

NPr = Cp * μ / k

Using the given values:
Cp = 0.583 J/(goC)
μ = 1936 lbm/(ft-h) (to convert from lbm/(ft-h) to J/(goC), divide it by 2.26)
k = 0.286 W/(moC) (to convert from W/(moC) to J/(goC), multiply it by 1000)

Therefore:
μ ≈ 1936 lbm/(ft-h) / 2.26 ≈ 857.52 lbm/(ft-h)
k ≈ 0.286 W/(moC) * 1000 ≈ 286 J/(goC)

Now we can calculate NPr:
NPr ≈ Cp * μ / k

NPr ≈ 0.583 J/(goC) * 857.52 lbm/(ft-h) / 286 J/(goC)

Simplifying this expression:
NPr ≈ 2.366 * 10^-3 lbm-ft/(ft-h)
NPr ≈ 2.366 * 10^-3

Thus, the estimated value of NPr is approximately 2.366 * 10^-3.

To determine the value of NPr with a calculator, we can use the exact values:

NPr = Cp * μ / k

Using the given values:
Cp = 0.583 J/(goC)
μ = 1936 lbm/(ft-h)
k = 0.286 W/(moC)

Now we can calculate NPr using a calculator or computer:

NPr = 0.583 * 1936 / 0.286

NPr ≈ 3955.561

Therefore, the value of NPr using a calculator is approximately 3955.561.

Expressing the result in scientific notation:

NPr = 3.955561 * 10^3

Thus, the value of NPr in scientific notation, with the correct number of significant figures, is approximately 3.956 * 10^3.

Answer 3

To estimate the value of NPr without using a calculator, we can use the given values and approximate calculations.

1. Let's start by converting the units of Cp and k to be consistent with each other. Since Cp is given in J/(g°C) and k is given in W/(m°C), we need to convert Cp to the same units as k.

Notice that 1 Joule (J) = 1 W·s, and 1 g = 0.001 kg. Therefore, 1 J/(g°C) = 1 W·s/(0.001 kg · °C).

We can convert Cp as follows:
Cp = 0.583 J/(g°C) = (0.583 W·s)/(0.001 kg · °C) = 583 W/(kg·°C).

2. Next, we need to convert the fluid viscosity (μ) from lbm/(ft-h) to the same units as Cp and k.

We can convert lbm to kg by multiplying by the conversion factor for lbm to kg (1 lbm = 0.4536 kg).
We can convert ft to m by multiplying by the conversion factor for ft to m (1 ft = 0.3048 m).
We can convert h to s by multiplying by the conversion factor for h to s (1 h = 3600 s).

As a result, μ can be converted as follows:
μ = 1936 lbm/(ft-h) = (1936 * 0.4536 kg)/(0.3048 m * 3600 s) = 0.826 kg/(m·s).

3. Finally, we can substitute the converted values of Cp, μ, and k into the formula for NPr, which is NPr = Cp·μ / k.

NPr = (583 W/(kg·°C))·(0.826 kg/(m·s)) / (0.286 W/(m·°C)).
NPr ≈ (583 W·kg·s/(kg·°C·m·s)) / (0.286 W·m/(m·°C·s)).
NPr ≈ (583 / 0.286) (kg·s / °C·m·°C).
NPr ≈ 2034.266 (kg·s / °C·m·°C).

The estimated value of NPr without using a calculator is approximately 2034.266 (kg·s / °C·m·°C).

Now, let's determine the value of NPr using a calculator to get a more precise result.

1. Substitute the original values directly into the formula: NPr = Cp·μ / k.

NPr = (0.583 J/(g°C)· 1936 lbm/(ft-h)) / (0.286 W/(moC)).
NPr ≈ (0.583 W·s/(0.001 kg · °C)· (1936 * 0.4536 kg)/(0.3048 m · 3600 s)) / (0.286 W/(m·°C)).
NPr ≈ (583 / 0.286) (kg·s / °C·m·°C).
NPr ≈ 2033.566 kg·s / °C·m·°C.

Therefore, the value of NPr, determined with a calculator, is approximately 2033.566 kg·s / °C·m·°C.

Both the estimated and calculated values of NPr should be expressed in scientific notation with the correct number of significant figures.